Summation of some infinite series by the methods of Hypergeometric functions and partial fractions

In this article we obtain the summations of some infinite series by partial fraction method and by using certain hypergeometric summation theorems of positive and negative unit arguments, Riemann Zeta functions, polygamma functions, lower case beta functions of one-variable and other associated functions. We also obtain some hypergeometric summation theorems for: 8F7[9/2, 3/2, 3/2, 3/2, 3/2, 3, 3, 1; 7/2, 7/2, 7/2, 7/2, 1/2, 2, 2; 1], 5F4[5/3, 4/3, 4/3, 1/3, 1/3; 2/3, 1, 2, 2; 1], 5F4[9/4, 5/2, 3/2, 1/2, 1/2; 5/4, 2, 3, 3; 1], 5F4[13/8, 5/4, 5/4, 1/4, 1/4; 5/8, 2, 2, 1; 1], 5F4[1/2, 1/2, 5/2, 5/2, 1; 3/2, 3/2, 7/2, 7/2; −1], 4F3[3/2, 3/2, 1, 1; 5/2, 5/2, 2; 1], 4F3[2/3, 1/3, 1, 1; 7/3, 5/3, 2; 1], 4F3[7/6, 5/6, 1, 1; 13/6, 11/6, 2; 1] and 4F3[1, 1, 1, 1; 3, 3, 3; −1].

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Some Inequalities for theq-Analogue of the Classical Riemann Zeta Functions and theq-Polygamma Functions

The Scientific World JOURNAL ◽

10.1155/2014/543593 ◽

2014 ◽

Vol 2014 ◽

pp. 1-4

Author(s):

Banyat Sroysang

Keyword(s):

Zeta Functions ◽

Polygamma Functions ◽

Riemann Zeta

We present the generalizations on some inequalities for theq-analogue of the classical Riemann zeta functions and theq-polygamma functions.

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Fibonacci-Zeta infinite series associated with the polygamma functions

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.4.95-103 ◽

2021 ◽

Vol 27 (4) ◽

pp. 95-103

Author(s):

Kunle Adegoke ◽

◽

Sourangshu Ghosh ◽

Keyword(s):

Infinite Series ◽

Fibonacci Numbers ◽

Linear Combinations ◽

Polygamma Functions ◽

Riemann Zeta

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

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XIV. On the convergence of infinite series of analytic functions

Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character ◽

10.1098/rsta.1905.0014 ◽

1905 ◽

Vol 204 (372-386) ◽

pp. 481-497

Keyword(s):

Analytic Function ◽

General Theory ◽

Infinite Series ◽

Analytic Functions ◽

Hypergeometric Functions ◽

Point Of View ◽

General Point ◽

Associated Functions ◽

Linear Differential ◽

Series Of Functions

In the first section of the following work an attempt is made to deal with the convergence of infinite series of functions defined by linear differential equations of the second order from the most general point of view. Functions of Lamé Bessel and Legendre are considered as examples. In the second section the results obtained are applied to the expansion of an arbitrary uniform analytic function of an arbitrary uniform analytic function of z in a series of hypergeometric functions, and the expansion is shown to be valid if the function is regular within a certain ellipse in the z -plane. An expansion in a series of Legendre’s associated functions is deduced by a transformation. The method has been applied by the writer to other cases, but the foregoing offer adequate illustration of the general theory.

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Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers

Applied Mathematics and Computation ◽

10.1016/s0096-3003(01)00172-2 ◽

2002 ◽

Vol 131 (2-3) ◽

pp. 593-605 ◽

Cited By ~ 25

Author(s):

Themistocles M. Rassias ◽

H.M. Srivastava

Keyword(s):

Infinite Series ◽

Harmonic Numbers ◽

Polygamma Functions ◽

Riemann Zeta ◽

Generalized Harmonic Numbers

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ANALOGUES OF A TRANSFORMATION FORMULA OF RAMANUJAN

International Journal of Number Theory ◽

10.1142/s179304211100454x ◽

2011 ◽

Vol 07 (05) ◽

pp. 1151-1172 ◽

Cited By ~ 7

Author(s):

ATUL DIXIT

Keyword(s):

Zeta Function ◽

Infinite Series ◽

Gamma Function ◽

Riemann Zeta Function ◽

Zeta Functions ◽

Transformation Formula ◽

Lost Notebook ◽

Riemann Zeta ◽

Limiting Case ◽

Special Case

We derive two new analogues of a transformation formula of Ramanujan involving the Gamma function and Riemann zeta function present in his Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular-type relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by A. P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.

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